The trajectory will not pass through the same point twice, but is not completely random. Lorenz attractors have been used to re-sample sequences in the following way: Imagine you have a sequence of musical notes. Pick a starting point on the Lorenz trajectory and associate each note with successive points. Now you have your notes laid out on the Lorenz attractor so that for any point in the space you can find the closest associated note. If you start on the Lorenz trajectory from a different point, you can sample the notes in a different sequence. This sample will be different from the original, but tends to preserve some of the structure. That is, the Lorenz attractor scrambles the sample, but in a chaotic way, not a random one.